When closed graph manifolds are finitely covered by surface bundles over \(S^1\)
A graph manifold is a 3-manifold constructed by identifying boundary components of compact Seifert-fibered 3-manifolds. From the authors' introduction: ``A classical fact is that Seifert manifolds with nonempty boundaries are finitely covered by surface bundles over \(S^1\). And a closed Seifert manifold is covered by a surface bundle over \(S^1\) if and only if its Euler number is zero. In the past several years, researchers have proven that some closed graph manifolds are not covered by surface bundles over \(S^1\) [\textit{J. Luecke} and \textit{Y. Q. Wu}, AMS/IP Stud. Adv. Math. 2 (pt. 1), 80-103 (1997; Zbl 0889.57024); \textit{W. D. Neumann}, Topology 36, No. 2, 355-378 (1997; Zbl 0872.57021)], and that graph manifolds with nonempty boundary are finitely covered by surface bundles over \(S^1\) [\textit{S. C. Wang} and \textit{F. C. Yu}, Math. Proc. Camb. Philos. Soc. 122, No. 3, 447-455 (1997; Zbl 0899.57011)]. In this paper we discuss the case when a closed graph manifold is finitely covered by a surface bundle over \(S^1\). For a given closed graph manifold \(M\), there are two associated matrices \(B(M)\) and \(W(M)\), where \(B(M)\) is defined in [\textit{S. C. Wang} and \textit{Y. Q. Wu}, Proc. Lond. Math. Soc., III. Ser. 68, No. 1, 203-224 (1994; Zbl 0841.57028)] and \(W(M)\) in [\textit{S. C. Wang} and \textit{F. C. Yu}, loc. cit. ]. We have two results. The first is Theorem 2.1, which is a necessary condition for \(M\) to be covered by a surface bundle. This condition can be read directly from the entries of \(B(M)\) and \(W(M)\), and is therefore very useful in practice. The second result is Theorem 3.1, which is a necessary and sufficient condition for \(M\) to be covered by a surface bundle. The condition is that a system of equations, whose coefficients are related to \(B(M)\) and \(W(M)\) in a complicated way, has a certain solution.'' This is not easy to compute in practice, in particular, it is difficult to show that the system does not have a required solution. Theorem 3.1 coincides substantially with the main result of [\textit{W. D. Neumann}, loc. cit.], but does not need the condition ``no self-pasting'' used there.
scientific article; zbMATH DE number 1279333
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